caps mathematics grade 11 sine, cosine and area rulesfnets.ws/caps/gr11...
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CAPS Mathematics
GRADE 11
Sine, Cosine
and Area Rules
4. Apply the Sine and the Cosine rules to solve
problems in 2-dimensions. Le ss on 4
1. Calculate the area of a triangle given an angle and the two
adjacent sides. Lesson 1
2. Apply the Sine Rule for triangles to calculate an unknown
side or an unknown angle of a given triangle. Le ss on 2
Outcomes for this Topic
3. Apply the Cosine Rule for triangles to calculate an unknown
side or an unknown angle of a given triangle. L e ss on 3
Lesson 1
The Area Rule
CAPS Mathematics
GRADE 11
sin opposite
hypotenuse
opposite hypotenuse
Trigonometric Ratios
In a , the 3 trigonometric ratios for an angle
are defined as follows:
right angled triangle
cos adjacent
hypotenuse
adjacent
tan opposite
adjacent
Some basic definitions – a reminder
Consider a non-right angled triangle ABC.
, and are the sides opposite angles , and respectively.
( This is the conventional way of labelling a triangle ).
a b c A B C
A B
C
b a
c N
h
1Area of base height
2
12
Area c h --- (1)
In ,ACN sin Ah
bsin b A h
Substituting for h in (1)
12
Area c sin b A12
sinArea bc A
The area formula of a triangle
Draw the perpendicular, , from to . h C BA
Any angle can be used as such in
area formula, so
12
sin Bca12
sinab C 12
sinbc AArea = = =
90
A similar argument gives the same formula for the area
if is obtuse i.e. B B
The formula always uses
2 sides and the
angle formed by those sides (Included )
A B
C
b
a
c N
h
90
Different forms of the area formula
c
b a a
C
B A
c
b a a
C
B A
c
b a a
C
B A
Three possible approaches to find
the area of a triangle
12
Area sin ab C
Any angle can be used in the formula, so
12
Area sin bc A
12
Area sin Bca
Find the area of PQR.
We know PQ and RQ so use the included angle Q
The area of a triangle – Example 1
Solution: We must use the angle formed by the
2 sides with the given lengths.
64
1Area of sin
2PQR QP QR Q
218 7 sin 64 cm
2
225,2 cm
r
B
A
C
r
21sin
2 Area r
Find the area of .ABCA useful application of the area formula occurs when we
hav 2 radiie a triangle formed by and of a a cho cirrd cle.
The area of a triangle – Example 2
1
Area sin2
CA CB C
But CA CB r
Find the areas of the triangles shown in the diagrams.
Give your answers accurate to 2 decimal digits
40
308 cm
10 cmradius 6 cm
120AOB
1) 2)
Classwork [Area of Triangles]
40
308 cm
10 cm
11) Area sin
2XYZ XY YZ Y
180 40 30 110Y
1sin
2z x Y
218 10 sin 110 cm
2
237,59 cm
Solution [Area of Triangles]
radius 6 cm120AOB
212) Area sin
2AOB r O
21
6 cm sin1202
215,59 cm
Given:
Solution [Area of Triangles]
Lesson 2
The Sine Rule
CAPS Mathematics
GRADE 11
One way to find unknown sides and angles in
is by using the :
non -right angled
triang Sine Rs ulele
The Sine Rule for Triangles
In ACN, sinh
Ab
sinh b A
Suppose is a scalene triangleABC
In , sinh
BCN Ba
sinh a B
sin sinb A a B
sin sinor
A B
a b
ab
c N
Drop CN AB
h
C
A B
b a
c
A
B C
b
a
c
h
Now sin sin
sin sin
h c B b C
B C
b c
can be turned so that is the base.
We then get
ABC BC
The Complete Sine Rule for Triangles
sin sin sinSo
A B C
a b c
When do we use the Sine Rule?The sine rule can be used in a triangle when:
Two angles and a side are given
Two sides and the non-included angle are given
To calculate second side
To calculate second angle
sin sinSolution: Use
A B
a b
sinsin
a BA
b10 sin 62
sin12
A
180 62 47,4 70,6 CThus
Application of the Sine Rule - Example 1
ABC A CIn , find the size of angles and .
47,4 A is opposite the shorter of the 2 given sides.
62 must be an acute angle.
(Only one possibility as can be seen from sketch)
A
A A
sin 5 sin 48sin
4
p QP
q
2
1
68,3
180 68,3 111,7
or
P
P
Application of the Sine Rule - Example 2
is opposite the longer of the 2 given sides.
48 can be an acute or obtuse angle.
( Two possibilities as can be seen from sketch below)
P
P P
1P
2Psin sin Use Solution :
Q P
q p
5, 4 48 .
.
PQR
QR PR Q
P
In it is given that:
and
Determine
13sin 55
sin 29
z
22,0 z
sin sin
z y
Z Y
In , find the length .XYZ XY
Application of the Sine Rule - Example 3
As the unknown is a side, we use the sine rule in
its reciprocal form. The unknown side is then at the top.
Solution :
sin
sin
y Zz
Y
2. In , find and the area of PQR QR PQR
1. In , find .
(Correct to two decimal places)
ABC B
Classwork [Sine Rule]
Find .
(2 decimal places)
B
10sin35sin
7B
1 55,02B
2or 180 55,02 124,98B
Given:
35 acute or obtuseB B
sin sin sin35 sin
7 10
A B B
a b
Solution
1
2
Obtained: 55,02
or 124,98
B
B
Given:
35 acute or obtuseB B
1 (Obtuse)B2 (Acute)B
Solution
2. Find and the area of .QR PQR
67
sin 64 sin80
QR
Given:
80R
67sin 64
sin80QR
61,15 cm
1Area of sin36
2PQR QP QR
167 61,15 sin36
2
21 204,09 cm
Solution
Lesson 3
The Cosine Rule
CAPS Mathematics
GRADE 11
2 2 2
2 2 2
2 cos
or
2 sin
b a c ac B
c a b ab C
c
b a a
C
B A
The Cosine Rule for is given by:ABC
The Cosine Rule for Triangles
2 2 2 2 cosa b c bc A
We use this form to find the third side when
two sides and included angle are given.
Symmetry also implies that:
Proof of the Cosine Rule
b a
In :CAD
2 2 2cos and x
A b x hb
In :BCD
22 2 2 2 22a h c x h c cx x
x c x
h
2 22 2 2cos2a c xb x c b A
2 2 2 cosb c bc A
Proofs for symmetrical results are similar.
A second form of the Cosine Rule 2 2 2 2 cos a b c bc AKnow:
2 2 22 cos bc A b c a 2 2 2
cos2
b c aA
bc
We use this form to find any angle of
a triangle when we know all 3 sides.
7 P R
Q
6
p
120
Find in the p PQR
Applications of the Cosine Rule - Example 1
Apply the Cosine Rule
2 2 2 2 cosp q r qr P
2 2 27 6 2 7 6 cos120p 127 11,3 1 decimal accuracyp
6
Y
Z 8
4
X
2 2 28 6 4cos
2(8)(6)
X
Find in the X XYZ
Solution: Use the Cosine Rule
Applications of the Cosine Rule - Example 2
29,0 ( 1 dec )X
2 2 2
cos2
y z xX
yz
2 2 2 2 cos c b a ba C
A
B C
c
30
Sine rule: sin sin
B C
b c
Find side and in the given . c B ABC
15b
19a
Cosine rule:
Applications of the Cosine Rule Example 3
2 2 215 19 2(15)(19)cos30 c
9,61 c ( 2 decimal places )
15 sin 30sin
9,61
B
51,3 B ( 1 dec. )
2. Find all the angles in , giving your
answers to one decimal place accuracy.
XYZ
1. Given with 6 cm; 4 cm and
60 . Find correct to 2 decimal digits.
ABC AB BC
ABC AC
Classwork
2 2 2 2 cosAC BC AB BC AB ABC
Given:Find (2 dec accuracy):AC
2 24 6 2 4 6 cos60
28
28 5,29 cmAC
Solution
sin sin sin 4sin 48.2Now sin
7
X Y y XY
x y x
Given:
Determine all angle measures of XYZ.
2 2 2
cos2
y z xX
yz
2 2 24 9 7Hence cos
2 4 9X
48,2X
25,2Y
Then 180 106,6Z X Y
Solution
Lesson 4
Basic Applications:
Problems in 2-D
CAPS Mathematics
GRADE 11
42 65
Problems in 2 dimensions: Example 1 1. Points and are in the same horizontal plane as ,
the foot of a vertical tower . 42 ; 65
and 25 . Calculate .
A B C
PC B PAC
AB m PC
P
CAB25 m
65 42 23BPA
25
sin 42 sin 23
AP
25sin 4242,81 m
sin 23AP
sin 65PC
AP
sin65 42,81sin65PC AP 38,8 m
23
Sine rule:
2. In the figure represents a proposed tunnel.
and are visible from a point .
The three points are in the same plane.
QR
Q R P
Q R
P
Given:
100 m; 60 m
and 110
QP PR
QPR
Calculate the length of tunnel.
100 60110
2 2 2100 60 2 100 60 cos110QR
133 mQR
Problems in 2 dimensions: Example 2
1. From the ends of a bridge , 101 metres long,
the angles of depression of a point on the ground
directly under the bridge is 42,2 and 70,1 .
Find the height, , of the bridge und
AB
P
h
er this point.
42,2 70,1A B101 m
P
h
Activity 1
42,2 70,1A B101 m
P
h
Question: Find h
180 42,2 70,1 67,7APB
101
sin 70,1 sin 67,7
AP
101 m sin 70,1
sin 67,7AP
102,65 mAP But sin 42,2102,65
h
102,65 sin 42,2 68,95 mh
Solution1
2. is a wall of a room, being the
line of the ceiling. is a picture rail,
with being directly below .
= 2 metres; and
2cos(a) Prove that
sin( )
(b)
ABCD AD
EF
E A
AE ACB x ECB y
xEC
x y
Find the length and height
of the wall if 33 and 20 .x y
2 m
xy
Activity 2
2 m
xy
Now and ACE x y CAD x
x
Hence, 90CAE x
From :AEC
2
sin 90 sin
EC
x x y
2sin 90
sin
xEC
x y
2cos
sin
x
x y
2(a) Prove that
2cos
sin( )
xEC
x y
Solution2
2 m
xy
Know:
2cos
sin
33 and 20
xEC
x y
x y
2cos337,46 m
sin13EC
cos cosBC
y BC EC yEC
Length of room 7,46 m cos20BC
7,01 m
Solution 2b